# Estimation of Riemannian distances between covariance operators and Gaussian processes

arXiv: https://arxiv.org/abs/2108.11683 · HTML: https://mlsanity.com/paper/2108.11683/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 48 references that are in the map.
- First posted on arXiv 26 Aug 2021.

## Builds on (most cited first, 6 of 6)

- [Functional Variational Bayesian Neural Networks](https://mlsanity.com/paper/1903.05779/) (2019)
- [On Sparse variational methods and the Kullback-Leibler divergence between stochastic processes](https://mlsanity.com/paper/1504.07027/) (2015)
- [Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes](https://mlsanity.com/paper/1801.01990/) (2018)
- [Bregman Divergences for Infinite Dimensional Covariance Matrices](https://mlsanity.com/paper/1403.4334/) (2014)
- [Entropic regularization of Wasserstein distance between infinite-dimensional Gaussian measures and Gaussian processes](https://mlsanity.com/paper/2011.07489/) (2020)
- [Convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings](https://mlsanity.com/paper/2101.01429/) (2021)

## Built on by (most cited first, 3 of 3)

- [Kullback-Leibler and Renyi divergences in reproducing kernel Hilbert space and Gaussian process settings](https://mlsanity.com/paper/2207.08406/) (2022)
- [Geometric Jensen-Shannon Divergence Between Gaussian Measures On Hilbert Space](https://mlsanity.com/paper/2506.10494/) (2025)
- [Generalized infinite dimensional Alpha-Procrustes based geometries](https://mlsanity.com/paper/2511.09801/) (2025)

## Abstract (arXiv)

In this work we study two Riemannian distances between infinite-dimensional positive definite Hilbert-Schmidt operators, namely affine-invariant Riemannian and Log-Hilbert-Schmidt distances, in the context of covariance operators associated with functional stochastic processes, in particular Gaussian processes. Our first main results show that both distances converge in the Hilbert-Schmidt norm. Using concentration results for Hilbert space-valued random variables, we then show that both distances can be consistently and efficiently estimated from (i) sample covariance operators, (ii) finite, normalized covariance matrices, and (iii) finite samples generated by the given processes, all with dimension-independent convergence. Our theoretical analysis exploits extensively the methodology of reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators. The theoretical formulation is illustrated with numerical experiments on covariance operators of Gaussian processes.